| L(s) = 1 | + (1.91 − 1.23i)2-s + (0.776 − 5.40i)3-s + (−1.16 + 2.55i)4-s + (−0.907 + 0.266i)5-s + (−5.16 − 11.3i)6-s + (8.82 + 10.1i)7-s + (3.50 + 24.3i)8-s + (−2.69 − 0.790i)9-s + (−1.41 + 1.62i)10-s + (−44.9 − 28.8i)11-s + (12.8 + 8.28i)12-s + (−21.9 + 25.3i)13-s + (29.4 + 8.65i)14-s + (0.735 + 5.11i)15-s + (22.0 + 25.4i)16-s + (−14.2 − 31.2i)17-s + ⋯ |
| L(s) = 1 | + (0.677 − 0.435i)2-s + (0.149 − 1.03i)3-s + (−0.145 + 0.318i)4-s + (−0.0812 + 0.0238i)5-s + (−0.351 − 0.770i)6-s + (0.476 + 0.550i)7-s + (0.154 + 1.07i)8-s + (−0.0997 − 0.0292i)9-s + (−0.0446 + 0.0515i)10-s + (−1.23 − 0.791i)11-s + (0.309 + 0.199i)12-s + (−0.468 + 0.541i)13-s + (0.562 + 0.165i)14-s + (0.0126 + 0.0880i)15-s + (0.344 + 0.397i)16-s + (−0.203 − 0.445i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.671 + 0.741i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.671 + 0.741i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.36130 - 0.603714i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.36130 - 0.603714i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 + (-88.8 + 65.3i)T \) |
| good | 2 | \( 1 + (-1.91 + 1.23i)T + (3.32 - 7.27i)T^{2} \) |
| 3 | \( 1 + (-0.776 + 5.40i)T + (-25.9 - 7.60i)T^{2} \) |
| 5 | \( 1 + (0.907 - 0.266i)T + (105. - 67.5i)T^{2} \) |
| 7 | \( 1 + (-8.82 - 10.1i)T + (-48.8 + 339. i)T^{2} \) |
| 11 | \( 1 + (44.9 + 28.8i)T + (552. + 1.21e3i)T^{2} \) |
| 13 | \( 1 + (21.9 - 25.3i)T + (-312. - 2.17e3i)T^{2} \) |
| 17 | \( 1 + (14.2 + 31.2i)T + (-3.21e3 + 3.71e3i)T^{2} \) |
| 19 | \( 1 + (-36.1 + 79.0i)T + (-4.49e3 - 5.18e3i)T^{2} \) |
| 29 | \( 1 + (-19.2 - 42.1i)T + (-1.59e4 + 1.84e4i)T^{2} \) |
| 31 | \( 1 + (-29.9 - 207. i)T + (-2.85e4 + 8.39e3i)T^{2} \) |
| 37 | \( 1 + (268. + 78.6i)T + (4.26e4 + 2.73e4i)T^{2} \) |
| 41 | \( 1 + (-315. + 92.7i)T + (5.79e4 - 3.72e4i)T^{2} \) |
| 43 | \( 1 + (48.6 - 338. i)T + (-7.62e4 - 2.23e4i)T^{2} \) |
| 47 | \( 1 - 460.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (159. + 183. i)T + (-2.11e4 + 1.47e5i)T^{2} \) |
| 59 | \( 1 + (78.4 - 90.5i)T + (-2.92e4 - 2.03e5i)T^{2} \) |
| 61 | \( 1 + (-57.8 - 402. i)T + (-2.17e5 + 6.39e4i)T^{2} \) |
| 67 | \( 1 + (-732. + 470. i)T + (1.24e5 - 2.73e5i)T^{2} \) |
| 71 | \( 1 + (-241. + 155. i)T + (1.48e5 - 3.25e5i)T^{2} \) |
| 73 | \( 1 + (379. - 831. i)T + (-2.54e5 - 2.93e5i)T^{2} \) |
| 79 | \( 1 + (-516. + 595. i)T + (-7.01e4 - 4.88e5i)T^{2} \) |
| 83 | \( 1 + (1.29e3 + 381. i)T + (4.81e5 + 3.09e5i)T^{2} \) |
| 89 | \( 1 + (105. - 736. i)T + (-6.76e5 - 1.98e5i)T^{2} \) |
| 97 | \( 1 + (-479. + 140. i)T + (7.67e5 - 4.93e5i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−17.55944402579243420474983237142, −15.85017867271983052651386568461, −14.14125503367680571539409953897, −13.25480191563905105324700492134, −12.27975886690477429599936335993, −11.12281920654267705144189524493, −8.646639268085941773992121489879, −7.36064352715927995335042758703, −5.09166933360828609623295624373, −2.58926151151444847432854054666,
4.14659688358253887156997723238, 5.35090018885185474126126455463, 7.57329911302301603393845055035, 9.768808743670697392230957846737, 10.54990249982423496562737260379, 12.71006266376815316926325078560, 14.02531521298387496015688098324, 15.24363671469252760158009300233, 15.69694376772737870468705050535, 17.29697537185432841707112662266
